I am a lifelong Oregonian and a backpacker for the past 40 years. I can assure you that in the PNW in a normal snow year, you will not find higher elevation trails that are clear of residual snow. There are signs that this year may not be a normal snow year, because the snow pack as of today is abnormally low, but there is no guarantee this situation will last to the end of spring.

You would be better off forming your plans around some other region. Colorado, or even New Mexico may well be more hospitable to backpackers in June. Also, the Sierra Nevada is experiencing very low snow accumulation as of right now, and it usually clears out earlier than Oregon of Washington, so it might be open at high elevation in June, too. It is very popular, so I'm not sure the best way to ensure you would get the permits you'd need. Our CA hikers might be able to fill you in on that head.